Does have an inverse function? Explain your answer.
Mathematics20271 markShort answer
Does $f(x)$ have an inverse function? Explain your answer.
Graph of f(x)
Answer
Answer
AI
No, $f(x)$ does not have an inverse function on its natural domain.
The graph fails the horizontal line test because a horizontal line intersects the curve at more than one point, so $f(x)$ is not one-to-one.
Algebraically, $y = (\sqrt{x})^2 - 6\sqrt{x} + 1 \Rightarrow (\sqrt{x})^2 - 6\sqrt{x} + (1 - y) = 0$.
Solving for $\sqrt{x}$: $\sqrt{x} = \frac{6 \pm \sqrt{36 - 4(1-y)}}{2} = 3 \pm \sqrt{y + 8}$, yielding two values of $x$ for each $y$ in the range.
